EigenvaluesofLévy CovariationMatrices Random matrix models for datasets with fixed time horizons Gregory Zitelli SIAM-ALA18, Hong Kong May, 2018
Laloux et al. (1999)
Which is real? Which is fake?
Contents 1. Motivation Markowitz portfolio theory, Bai’s Theorem 2. M–P Law Universality, eigenvalue outliers 3. Sample Lévy Covariance Ensemble (SLCE) Lévy processes, GGC/EGGC processes 4. Limiting Spectral Distributions Stieltjes transform algorithm, approximate densities
Motivation
Markowitz Portfolio Theory p risky assets with return vector r = [ r 1 . . . r p ] . µ = E [ r ] is the fixed 1 × p vector of expected returns µ µ Σ is the symmetric p × p covariance matrix. Σ Σ w is the 1 × p vector of weights on each asset, w1 † ≤ 1 . For fixed 0 < σ 2 < ∞ , consider the optimization µ † (expected return on portfolio) maximize µ w µ Σ w † ≤ σ 2 (acceptable volatility) w ∈ R 1 × p , w1 † ≤ 1 , w Σ subject to Σ Let R ( σ 2 ,µ Σ) denote the optimal return µ = w µ µ † . µ µ, Σ Σ µ 1/29
Markowitz Portfolio Theory µ † = R ( σ 2 ,µ maximize µ, Σ Σ Σ) w µ µ µ w ∈ R 1 × p , w1 † ≤ 1 , w Σ Σ w † ≤ σ 2 subject to Σ Σ) and the optimal w have closed-form R ( σ 2 ,µ µ, Σ Σ µ expressions in terms of σ 2 , µ µ , and Σ Σ . Σ µ In practice, µ µ and Σ Σ are unknown. µ Σ How accurate is the portfolio construction when µ µ and Σ µ Σ Σ are estimated by independent sampling? 2/29
Estimating the Efgicient Frontier 3/29
Markowitz Portfolio Theory Rectangular N × p data matrix. Let λ = p / N ∈ (0 , 1) , N samples, p assets. Bouchaud and Potters, mid 2000s, out-of-sample risk √ underestimated by 1 − λ , unproven. 2006, Bai Zhidong, Wong Wing-Keung, A Note on the Mean-Variance Analysis of Self-Financing Portfolios (unpublished), full proof published in 2009. Overestimation of returns, underestimation of risk. Relies on Marčenko–Pastur law. 4/29
lim Bai’s Theorem p assets, N observations, p / N → λ (0 , 1) . Σ N sequence of p × p covariance matrices Σ Σ µ N sequence of 1 × p expected return vectors. µ µ Σ 1/2 Model: X N = 1 N . 1 † µ 1 µ N + Y N Σ Σ µ Under mild assumptions (finite fourth moment), √ µ N , � R ( σ 2 , � Σ µ µ Σ Σ N ) 1 Σ N ) = R ( σ 2 ,µ µ µ N , Σ Σ 1 − λ N →∞ Proof is built on the Marčenko–Pastur (M–P) law applied to the Σ 1/2 sample data matrix Y N Σ N . Σ 5/29
M–PLaw
p Marčenko–Pastur Law · · · y 1 , 1 y 1 , p . . ... . . . . y j , k i.i.d. ∼ Y Y = N , . . for all N . . . . · · · y N , 1 y N , p � �� � Rows are independent samples of p i.i.d. random variables with true covariance I p , sample covariance matrix is S = 1 N Y † Y If p is fixed and N → ∞ , S → I p , S has p eigenvalues converging to 1 . What if p , N → ∞ , p / N → λ ∈ (0 , 1) ? 6/29
Marčenko–Pastur Densities 7/29
Example N = 2000 , p = 500 , λ = 0 . 25 , i.i.d. Gaussian entries. 8/29
Laloux et al. (1999) 9/29
Outliers? N = 2000 , p = 500 , λ = 0 . 25 , i.i.d. Lognormal entries (normalized). 10/29
SampleLévyCovariance Ensemble(SLCE)
N N N → ∞ Marčenko–Pastur: [ y 1 , k y 2 , k y 3 , k . . . y N − 1 , k y N , k ] † � ↘ N ′ [ y 1 , k y 2 , k y 3 , k . . . y N − 1 , k y N , k y N +1 , k . . . y N ′ − 1 , k y N ′ , k ] † New approach, fixed horizon [0 , T ] : y N , k ] † [ y 1 , k . . . y 2 , k y 3 , k y N − 1 , k � � � � N ′ y N ′ − 1 , k y N ′ , k ] † [ y 1 , k y 2 , k y 3 , k . . . . . . y N − 1 , k y N , k y N +1 , k 11/29
Marčenko–Pastur 12/29
Marčenko–Pastur, N increases 12/29
Fixed time horizon 13/29
Fixed time horizon, N increases 13/29
Lévy Processes Correspondence between Lévy processes X t ⇐ ⇒ infinitely divisible (ID) distributions X through X 1 ∼ X Lévy–Khintchine: Decomposition of characteristic function in terms of Brownian motion (continuous part) and an additional measure Π . 14/29
Lévy Processes [ ] ∫ 1 t log φ X t ( ϑ ) = i µϑ − 1 i ϑ x 2 σ 2 ϑ 2 + e i ϑ x − 1 − d Π( x ) 1 + x 2 R Π positive Borel measure, Π( { 0 } ) = 0 d Π( x ) integrable near ±∞ x 2 d Π( x ) integrable near zero behavior of d Π( x ) near 0 ⇐ ⇒ path properties of X t tail behavior of d Π( x ) ⇐ ⇒ tail behavior of f X 15/29
GGCs Thorin (1977): Nonnegative Lévy process such that d Π( x ) = x − 1 g + ( x ) dx , x > 0 where g + : (0 , ∞ ) → R is completely monotone (CM). Extended GGC (EGGC): Lefu and right tails, { x − 1 g + ( x ) dx , x > 0 d Π( x ) = | x | − 1 g − ( | x | ) dx , x < 0 g ± both CM. 16/29
EGGCs Good properties: infinite activity, continuous densities. Lots of common distributions are GGC or EGGC. α -stable, Student’s t , lognormal, gamma, Laplace, Pareto, generalized skew hyperbolic, generalized inverse Gaussian. Popular models of asset returns are EGGC. variance-gamma (VG) model of Madan and Seneta (1990), normal-inverse Gaussian (NIG) model of Barndorfg-Nielsen (1997), CGMY model of Carr et al. (2002). Symmetric EGGCs are precisely the processes created by Brownian motion time-changed by a GGC! 17/29
EGGCs 18/29
p Sample Lévy Covariance Ensemble · · · x 1 , 1 x 1 , p . . ... . . . . x j , k i.i.d. ∼ X T / N X = N , . . for each N . . . . · · · x N , 1 x N , p � �� � Fix an EGGC process X t . Fix a time horizon T > 0 . Fix a shape parameter λ ∈ (0 , 1) , p : N → N with p ( N )/ N → λ . Let X = X ( N ) be a sequence of independent random matrices with i.i.d. entries, such that [ X ( N )] jk ∼ X T / N . 19/29
p Sample Lévy Covariance Ensemble · · · x 1 , 1 x 1 , p . . ... . . . . x j , k i.i.d. ∼ X T / N X = N , . . for each N . . . . · · · x N , 1 x N , p � �� � Now consider the sample covariance S = 1 T X † X and its eigenvalues. Are these distinct from M–P? 20/29
Variance-Gamma Process 21/29
Normal-Inverse-Gaussian Process 22/29
LimitingSpectralDistributions
N N Variance of columns MP law relies on the convergence of the sample variance of the columns to converge to something non-random. · · · x 1 , 1 x 1 , p . . ... . . . . . . . . . . · · · x N , 1 x N , p ∑ 1 x 2 j , k j =1 If x j , k ∼ Y (mean zero) are i.i.d. then these converge to Var ( Y ) . If x j , k ∼ X T / N then these converge to [ X ] T , the (random) quadratic variation process corresponding to X t at time t = T . 23/29
Random column variance √ ν 1 · · · · · · 0 . . . 0 x 1 , 1 x 1 , p z 1 , 1 z 1 , p . . . . . ... ... . . . . √ ν 2 . . . . . . 0 ∼ . . . . . ... . . . . . . . . . . 0 √ ν p · · · · · · 0 · · · 0 x N , 1 x N , p z N , 1 z N , p ν j ∼ [ X ] T are i.i.d. copies of the quadratic variation process. This type of matrix has been well-studied, limiting spectral densities can be approximated by a modification of the algorithm introduced in Yao et al. (2015). 24/29
w z Algorithm ∫ Let s ( z ) = x − z dL ( x ) be the Stieltjes transform of the 1 R limiting spectral distribution L for z ∈ C + . The modified Stieltjes transform is s ( z ) = − 1 − λ + λ s ( z ) . If ν ∼ [ X ] T is the measure of the quadratic variation process at time T > 0 , then s ( z ) is given by 1 ∫ s ( z ) = − z + λ 1+ ws ( z ) d ν ( w ) R Iterations of the map for z = x + i ϵ can be used with Stieltjes inversion to approximate the density for L . Density is approximately f ( x ) ≈ s ( x + i ϵ ) . 25/29
Estimating the Density 26/29
Empirical deviations from M–P Comparison of S&P 500 (lefu, 476 assets) and Nikkei 225 (middle, 221 assets). Daily data (top) June 2013-May 2017 (approx. 900 datapoints) and minute-by-minute (bottom) January 2017-May 2017 (approx. 40000 [SPX] and 30000 [NKY] datapoints). Right column: SLCE generated with i.i.d. NIG entries, T = 0 . 05 × 900 (top) and T = 0 . 05 × 100 (bottom). 27/29
Empirical deviations from M–P 28/29
Conclusions
Conclusions Empirical evidence shows limiting distributions are independent of odd cumulants (skewness). Sensitive to higher moments, kurtosis not enough to determine shape (VG and NIG were normalized to have same excess kurtosis). Behavior of eigenvectors and top eigenvalue need to be investigated, efgects accuracy of weights in portfolio selection. 29/29
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